By using the quantum kinetic equation for electrons and considering the electron - optical phonon interaction, we obtain analytical expressions for the Hall conduc[r]
VNU Journal of Mathematics – Physics, Vol 29, No (2013) 33-43 Calculation of the Hall Coefficient in Doped Semiconductor Superlattices with a Perpendicular Magnetic Field under the Influence of a Laser Radiation Bui Dinh Hoi*, Pham Thi Trang, Nguyen Quang Bau Faculty of Physics, VNU University of Science, 334 Nguyen Trai, Hanoi, Vietnam Received 20 February 2013 Revised 03 March 2013; accepted 25 March 2013 Abstract: We consider a model of the Hall effect when a doped semiconductor superlattice (DSSL) with a periodical superlattice potential in the z-direction is subjected to a crossed dc electric field (EF) E = E ,0 ,0 and magnetic field B = ,0 ,B , in the presence of a laser radiation characterized by electric field E = ,E sin Ωt ,0 (where E and Ω are the ( ) ( ( ) ( ) ) amplitude and the frequency of the laser radiation, respectively) By using the quantum kinetic equation for electrons and considering the electron - optical phonon interaction, we obtain analytical expressions for the Hall conductivity as well as the Hall coefficient (HC) with a dependence on B, E , E , Ω , the temperature T of the system and the characteristic parameters of DSSL The analytical results are computationally evaluated and graphically plotted for the GaAs:Si/GaAs:Be DSSL Numerical results show the saturation of the HC as the magnetic field or the laser radiation frequency increases This behavior is similar to the case of low temperature in two-dimensional electron systems Keywords: Hall effect, Quantum kinetic equation, Doped superlattices, Parabolic quantum wells, Electron - phonon interaction Introduction∗ It is well-known that the confinement of electrons in low-dimensional systems (nanostructures) makes their properties different considerably in comparison to bulk materials [1, 2, 3], especially, the optical and electrical properties become extremly unusual This brings a vast possibility in application to design optoelectronics devices In the past few years, there have been many papers dealing with problems related to the incidence of electromagnetic wave (EMW) in low-dimensional semiconductor systems The linear absorption of a weak EMW caused by confined electrons in low dimensional _ ∗ Corresponding author ĐT: 84-989343494 E-mail: hoibd@nuce.edu.vn 33 34 B.D Hoi et al / VNU Journal of Mathematics-Physics, Vol 29, No (2013) 33-43 systems has been investigated by using Kubo - Mori method [4, 5] Calculations of the nonlinear absorption coefficients of a strong EMW by using the quantum kinetic equation for electrons in bulk semiconductors [6], in compositional semiconductor superlattices [7, 8] and in quantum wires [9] have also been reported Also, the Hall effect in bulk semiconductors in the presence of EMW has been studied in much details by using quantum kinetic equation method [10 - 14] In Refs 10 and 11 the odd magnetoresistance was calculated when the nonlinear semiconductors are subjected to a magnetic field and an EMW with low frequency, the nonlinearity is resulted from the nonparabolicity of distribution functions of carriers In Refs 12 and 13, the magnetoresistance was derived in the presence of a strong EMW for two cases: the magnetic field vector and the electric field vector of the EMW are perpendicular [12], and are parallel [13] The existence of the odd magnetoresistance was explained by the influence of the strong EMW on the probability of collision, i.e., the collision integral depends on the amplitude and frequency of the EMW This problem is also studied in the presence of both low frequency and high frequency EMW [14] Moreover, the dependence of magnetoresistance as well as magnetoconductivity on the relative angle of applied fields have been considered carefully [10 - 14] The behaviors of this effect are much more interesting in low-dimensional systems, especially two-dimensional electron gas (2DEG) system To our knowledge, the Hall effect in low-dimensional semiconductor systems in the presence of an EMW remains a problem to study Therefore, in this work, by using the quantum kinetic equation method we study the Hall effect in a doped semiconductor superlattice (DSSL) with the superlattice potential assumed to be in the z-direction, subjected to a crossed dc electric field (EF) E = E ,0 ,0 and magnetic field B = ,0 ,B ( B is applied perpendicularly to the plane of free motion of electrons ( ( ) ) - the (x-y) plane, so we temporarily call the perpendicular Hall coefficient (PHC) in this study), in the presence of an EMW characterized by electric field E = ,E sin Ωt ,0 We only consider the case ( ( ) ) of high temperatures when the electron - optical phonon interaction is assumed to be dominant and electron gas is nondegenerate We derive analytical expressions for the Hall conductivity tensor and the PHC taking account of arbitrary transitions between the energy levels The analytical result is numerically evaluated and graphed for the GaAs:Si/GaAs:Be DSSL to show clearly the dependence of the PHC on above parameters The present paper is organized as follows In the next section, we show briefly the analytical results of the calculation Numerical results and discussion are given in Sec Finally, remarks and conclusions are shown briefly in Sec Hall effect in a parabolic quantum well under the influence of a laser radiation 2.1 Quantum kinetic equation for electrons We consider a simple model of a DSSL (n-i-p-i superlattice) in which electron gas is confined by an additional potential along the z-direction and free in the (x-y) plane The motion of an electron is confined in each layer of the system and that its energy spectrum is quantized into discrete levels in B.D Hoi et al / VNU Journal of Mathematics-Physics, Vol 29, No (2013) 33-43 35 the z-direction If the DSSL is subjected to a crossed dc EF E = E ,0 ,0 and magnetic field B = ,0 ,B , also a laser radiation (strong EMW) is applied in the z-direction with the electric field vector E = ,E sin Ωt ,0 , then the Hamiltonian of the electron-optical phonon system in the above ( ( ) ) ( ( ) ) mentioned regime in the second quantization representation can be written as H = H0 + U , (1) e k − A ( t ) aN+ ,n , k aN ,n , k + ∑ ωq bq+ bq , N ,n y y y c q N ,n , k y U= ∑ ∑∑ DN ,n , N ',n ' ( q ) aN+ ',n ' k + q aN ,n ,ky bq + b−+q , H0 = ∑ε N , N ' n ,n ' q , k y y y ( ) (2) (3) are electron states before and after scattering; k y = (0, k y ,0) ; ωq is the energy of an optical phonon with the wave vector q = (qx , q y , qz ) ; aN+ ,n ,k and aN ,n ,k ( bq+ y y and bq ) are the creation and annihilation operators of electron (phonon), respectively; A ( t ) is the vector potential of the EMW; and DN ,n , N ',n ' ( q ) is the matrix element of interaction which depends on where N ,n,k y and N ',n',k y + q y the initial and final states of electron and the interacting mechanism In our model, the DSSL potential can be considered as a multiquantum-well structure with the parabolic potential in each well, and if we neglect the overlap between the wave functions of adjacent wells, the single-particle wave function and corresponding eigenenergy of an electron in a single potential well are given by [16 - 18] ik y Φ N ( x − x0 ) e y Φ n ( z ) , Ly N ,n,k y = 1 ε N ,n k y = N + ωc + ε n − vd k y + mvd2 ; 2 ( ) N ,n = 0,1, , (4) (5) where N is the Landau level index and n being the subband index; Ly is the normalization length in the y direction; ωc = eB / m being the cyclotron frequency and vd = E1 / B is the drift velocity of electron Also, Φ N represents harmonic oscillator wave functions, centered at x0 = − 2B ( k y − mvd / ) where B = / ( mωc ) is the radius of the Landau orbit in the (x-y) plane Φ n ( z ) and ε n are the wave functions and the subband energy values due to the superlattice confinement potential in the zdirection, respectively, given by Φn ( z ) = z2 z exp − Hn , 2n n ! π z 2 z z 1 ε n = n + ω p , (6) (7) 36 B.D Hoi et al / VNU Journal of Mathematics-Physics, Vol 29, No (2013) 33-43 with H n ( z ) being the Hermite polynomial of nth order and z = / ( mωp ) , ωp is the plasma frequency characterizing for the DSSL confinement in the given z-direction, by 1/ ωp = ( 4π e nD / κ m ) , where κ is the electronic constant (vacuum permittivity) and nD is the doped concentration The matrix element of interaction, DN ,n , N ',n ' ( q ) is given by [16, 17] DN , n, N ',n ' ( q ) = Cq 2 I n ,n ' ( ± qz ) J N , N ' ( u ) , (8) where Cq is the electron-phonon interaction constant which depends on scattering mechanism, for electron - optical phonon interaction [6, 7, 16, 17] Cq = 2π e ω0 ( χ ∞−1 − χ 0−1 ) / ( κ 0V0 q ) , where V0 is the normalization volume of specimen, χ and χ ∞ are the static and the high-frequency dielectric constants, respectively; I n ,n ' ( ± qz ) is the form factor of electron, given by s0 d I n ,n ' ( ± qz ) = ∑ ∫ e± iqz d φn ( z −℘d ) φn ' ( z −℘d )dz , (9) ℘=1 with is d the period and s0 J N , N ' (u ) = ( N '!/ N !) e− u u N '− N LNN '− N (u ) u = 2B q⊥2 , q⊥2 = qx2 + q y2 is the number of periods of the DSSL; also with LNM ( x) is the associated Laguerre polynomial, By using Hamiltonian (1) and the procedures as in the previous works [10 - 14], we obtain the quantum kinetic equation for electrons in the single (constant) scattering time approximation f N ,n ,k − f 2π ∂f N ,n ,ky k y ∂f N ,n ,ky +∞ λ y + − eE1 + ωc k y ∧ h + D q Js ( =− ∑ ∑ N ,n, N ',n ' ) s∑ τ m ∂r N ',n ' q ∂k y Ω =−∞ × f N ',n ',k + q ( N q + 1) − f N ,n ,k N q δ ε N ',n ' k y + q y − ε N ,n k y − ωq − sΩ y y y + f N ',n ',k − q N q − f N ,n ,k ( N q + 1) δ ε N ',n ' k y − q y − ε N ,n k y + ωq − sΩ , (10) y y y where h = B / B is the unit vector along the magnetic field; the notation ‘ ∧ ’ represents the cross product (or vector product); f is the equilibrium electron distribution function (Fermi - Dirac distribution); f N ,n, k is an unknown electron distribution function perturbed due to the external fields; ( ) { ( ( ( ( ) ) ( ) ( ) ) )} y τ is the electron momentum relaxation time, which is assumed to be constant; f N ,n, k ( N q ) is the y time-independent component of the distribution function of electrons (phonons); J s ( x) is the s th order Bessel function of argument x; δ ( ) being the Dirac's delta function; and λ = eE0 q y / (mΩ) Equation (10) is fairly general and can be applied for any mechanism of interaction It was obtained in both bulk semiconductors and compositional superlattices [10 - 14] In the following, we will use this expression to derive the Hall conductivity tensor as well as the PHC 37 B.D Hoi et al / VNU Journal of Mathematics-Physics, Vol 29, No (2013) 33-43 2.2 Expressions for the Hall conductivity and the Hall coefficient For simplicity, we limit the problem to the cases of s = −1,0,1 This means that the processes with e more than one photon are ignored If we multiply both sides of Eq (10) by k yδ ε − ε N ,n k y and m carry out the summation over N and k y , we have the equation for the partial current density jN ,n, N ',n ' ( ε ) (the current caused by electrons that have energy of ε): ( jN ,n , N ',n ' ( ε ) τ ( )) + ωc h ∧ jN ,n , N ',n ' ( ε ) = QN ,n ( ε ) + S N ,n , N ',n ' ( ε ) , (11) where ∂f N ,n ,ky e QN ,n ( ε ) = − ∑ k y F m N ,n ,k y ∂k y δ(ε − ε N ,n (k y )) , F = eE1 (12) and λ2 2πe − f k f S N ,n ,N ',n' ( ε ) = D q N − ( ) ∑ ∑ ∑ N ,n ,N ',n' q y N ,n ,k y N ',n',k y + qy m ky ,q N ',n' N ,n 2Ω λ2 λ2 + Ω + ×δ ε N ',n' k y + q y − ε N ,n k y − ωq + δ ε k + q − ε k − ω N ',n' y y N ,n y q 4Ω 4Ω ×δ ε N ',n' k y + q y − ε N ,n k y − ωq − Ω λ + f N ',n',k − q − f N ,n ,k − δ ε N ',n' k y − q y − ε N ,n k y + ωq y y y 2Ω λ λ2 + Ω + + δ ε k − q − ε k + ω δ ε N ',n' k y − q y − ε N ,n k y − ωq − Ω N ',n' y y N ,n y q 2 4Ω 4Ω ×δ ε − ε N ,n k y ( ( ( ( ) ) ) ( ) ( ) ( ( ) ( ) ( ) ( ) ) ) ) ( ( ( ( ) ( ) ) ( ( ) ( ) ) ( )) (13) Solving (11) we have the expression for jN ,n, N ',n ' ( ε ) as follow jN ,n, N ',n ' ( ε ) = τ Q (ε ) + S N ,n , N ',n ' (ε ) ) − ωcτ [h ∧ QN ,n (ε )] + [ h ∧ S N ,n, N ',n ' (ε )] 2 ( N ,n + ωc τ { 2 c +ω τ ( ( QN ,n (ε )h + S N ,n , N ',n ' (ε )h h The total current density is given by )} ) (14) 38 B.D Hoi et al / VNU Journal of Mathematics-Physics, Vol 29, No (2013) 33-43 ∞ J = ∫ jN ,n , N ', n ' ( ε )d ε or J i = σ im E1m (15) Inserting (14) into (15) we obtain the expressions for the Hall current J i as well as the Hall conductivity σ im after carrying out the analytical calculation To this, we consider only the electron-optical phonon interaction at high temperatures, the electrons system is nondegenerate and assumed to obey the Boltzmann distribution function in this case Also, we assume that phonons are dispersionless, i.e, ωq ≈ ω0 , N q ≈ N = k BT / (ω0 ) , where ω0 is the frequency of the longitudinal optical phonons, assumed to be constant, k B being Boltzmann constant Otherwise, the summations over k y and q are changed into the integrals as follows ∑ ( ) → ky ∑ ( ) → q V0 4π Ly Lx /2 2B ∫ ( )dk y , +π / d +∞ ∫ (16) − Lx / 2B ( )q⊥ dq⊥ ∫ dqz = −π / d V0 4π 2B +π / d +∞ ∫ ( )du ∫ (17) dqz , −π / d here, Lx is the normalization length in the x-direction After some manipulation, we obtain the expression for the conductivity tensor: −1 e2 τ + ωc2 τ2 ) ( δij − ωc τε ijk hk + ωc2 τ2 hi h j ) aδ jm + bδ j δm − ωc τε mp hp + ωc2 τ h hm , (18) ( where δ ij is the Kronecker delta; ε ijk being the antisymmetric Levi - Civita tensor; the Latin { σim = } symbols i, j , k , l , m, p stand for the components x, y, z of the Cartesian coordinates; a=− β vd Ly I 2π m ∑e β ( ε F − ε N ,n ) , (19) N ,n with ε F is the Fermi level; and b= β AN Ly I τ ∑ ∑ I ( n, n '){b1 + b2 + b3 + b4 + b5 + b6 + b7 + b8 }, 2 8π m + ωc2τ N ,n N ', n ' b1 = b2 = − ( N + M ) ! eBξ δ ( X1 ) , exp β (ε F − ε N ,n ) M N! ( N + M ) ! δ ( X1 ) , exp β (ε F − ε N ,n ) 2M N! θ eBξ ( N + M ) ! b3 = δ ( X2 ), exp β (ε F − ε N ,n ) 4M N! θ eBξ ( N + M ) ! b4 = δ ( X3 ), exp β (ε F − ε N , n ) 4M N! θ eBξ (20) B.D Hoi et al / VNU Journal of Mathematics-Physics, Vol 29, No (2013) 33-43 39 eBξ N! b5 = δ ( X4 ), exp β (ε F − ε N ,n ) M ( N + M ) ! N! b6 = − δ ( X4 ), exp β (ε F − ε N ,n ) 2M ( N + M ) ! θ eBξ N! b7 = δ ( X5 ), exp β (ε F − ε N , n ) 4M ( N + M ) ! θ eBξ N! b8 = δ ( X6 ), exp β (ε F − ε N ,n ) 4M ( N + M ) ! θ eBξ X = ( N '− N )ωc + (n '− n)ωp − eE1ξ − ω0 , X = X + Ω , X = X − Ω , X = ( N − N ')ωc + (n '− n)ωp + eE1ξ + ω0 , X = X + Ω , X = X − Ω , M =| N − N ' |= 1,2,3, , ξ= ( α = vd , ) θ = e E02 / ( m Ω ) , β = / ( k BT ) , N +1/ + N +1+1/ B / , −1 A = 2π e ω0 ( χ ∞−1 − χ 0−1 ) / κ , 1 1 ε N ,n = N + ωc + n + ωp + mvd2 , 2 2 −2 I = a1 (αβ ) exp ( αβ a1 ) + exp ( −αβ a1 ) − ( αβ ) exp (αβ a1 ) − exp ( −αβ a1 ) , we have set +π / d I (n, n ') = ∫ π I n, n ' ( ± qz ) dqz a1 = Lx / 2 2B ; where (21) − /d which will be numerically evaluated by a computational program The divergence of delta functions is avoided by replacing them by the Lorentzians as [19] δ (X ) = 1 Γ 2 π X +Γ (22) where Γ is the damping factor associated with the momentum relaxation time τ by Γ = / τ The appearance of the parameter ξ is due to the replacement of q y by eBξ / , where ξ is a constant of the order of B The purpose is to a simplicity in performing the integral over q⊥ This has been done in [16] and is equivalent to assuming an effective phonon momentum evd q y ≈ eE1ξ The PHC is given by the formula [20] RH = − where σ yx and σ xx are given by Eq (18) σ yx , B σ 2xx + σ 2yx (23) 40 B.D Hoi et al / VNU Journal of Mathematics-Physics, Vol 29, No (2013) 33-43 Equations (18) and (23) show the complicated dependences of the Hall conductivity tensor and the PHC on the external fields, including the EMW It is obtained for arbitrary values of the indices N, n, N′ and n′ However, it contains the term I ( n,n' ) which is difficult to find out the exact analytical result due to the presence of the Hermite polynomials We will numerically evaluatethis term by the computational method In the next section, we will give a deeper insight into these results by carrying out a numerical evaluation and a graphic consideration by the computational method Numerical results and discussion In this section we present detailed numerical calculations of the Hall conductivity and the PHC in a DSSL subjected to the uniform crossed magnetic and electric fields in the presence of a strong EMW For the numerical evaluation, we consider the n-i-p-i superlattice of GaAs:Si/GaAs:Be with the parameters [15, 17]: ε F = 50meV , χ ∞ = 10.9 , χ = 12.9 , ω0 = 36.25meV , and m=0.067 × m0 (m0 is mass of a free electron) For the sake of simplicity, we also choose N = 0, N ' = 1, n = 0, n ' = ÷ (the lowest and the first-excited levels), τ = 10−12 s, Lx = Ly = 10−9 m and the number of periods used in the computation is 30 Figure Hall coefficient (arb Units) as functions of the EMW frequency at different at B = 4.00T (solid line), B = 4.05T (dashed line), and B = 4.10T (dotted line) Here, E1 = × 105 Vm −1 , E0 = 105 Vm −1 , Figure Hall coefficient (arb Units) as functions of the magnetic field B at different values of the superlattice period: d = 15nm (solid line), d = 16nm (dashed line), and d = 17nm (dotted line) Here, ωp = × 1013 s −1 , d = 15nm , and T = 270K E1 = × 105 Vm −1 , E0 = 105 Vm −1 , ωp = × 1013 s −1 , Ω = × 1013 s −1 , and T = 270K B.D Hoi et al / VNU Journal of Mathematics-Physics, Vol 29, No (2013) 33-43 41 Figure Hall coefficient (arb Units) as functions of the doped concentration at temperature of 240K (solid line), 270K (dashed line), and 300K (dotted line) Here B = 4T , E1 = × 105 Vm −1 , E0 = 105 Vm −1 , ωp = × 1013 s −1 , d = 15nm , and Ω = × 1013 s −1 Figure shows the PHC as a function of the EMW frequency at different values of the magnetic field In the region Ω